Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.9% → 98.3%

Time: 9.1s

Alternatives: 13

Speedup: 1.0×

• 25.0% 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: 92.9% 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: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{-41}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.85e-41)
   (+ x (* z (/ (- y x) t)))
   (if (<= t 7.2e-23) (+ x (/ (* z (- y x)) t)) (+ x (/ z (/ t (- y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.85e-41) {
		tmp = x + (z * ((y - x) / t));
	} else if (t <= 7.2e-23) {
		tmp = x + ((z * (y - x)) / t);
	} else {
		tmp = x + (z / (t / (y - 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 <= (-3.85d-41)) then
        tmp = x + (z * ((y - x) / t))
    else if (t <= 7.2d-23) then
        tmp = x + ((z * (y - x)) / t)
    else
        tmp = x + (z / (t / (y - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.85e-41) {
		tmp = x + (z * ((y - x) / t));
	} else if (t <= 7.2e-23) {
		tmp = x + ((z * (y - x)) / t);
	} else {
		tmp = x + (z / (t / (y - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.85e-41:
		tmp = x + (z * ((y - x) / t))
	elif t <= 7.2e-23:
		tmp = x + ((z * (y - x)) / t)
	else:
		tmp = x + (z / (t / (y - x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.85e-41)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	elseif (t <= 7.2e-23)
		tmp = Float64(x + Float64(Float64(z * Float64(y - x)) / t));
	else
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.85e-41)
		tmp = x + (z * ((y - x) / t));
	elseif (t <= 7.2e-23)
		tmp = x + ((z * (y - x)) / t);
	else
		tmp = x + (z / (t / (y - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.85e-41], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.2e-23], N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8499999999999999e-41

   1. Initial program 88.0%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   if -3.8499999999999999e-41 < t < 7.1999999999999996e-23

   1. Initial program 98.4%

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

   if 7.1999999999999996e-23 < t

   1. Initial program 82.4%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.85 \cdot 10^{-41}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 2: 97.4% 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 91.5%

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

   1. +-commutative 91.5%

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

   2. *-commutative 91.5%

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

   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: 54.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;t \leq -9 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-93}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) y)))
   (if (<= t -9e-27)
     x
     (if (<= t -2.5e-306)
       t_1
       (if (<= t 2.8e-252)
         (/ (* z (- x)) t)
         (if (<= t 4.1e-184)
           t_1
           (if (<= t 1.15e-93)
             (* z (/ (- x) t))
             (if (<= t 1.45e+33) t_1 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if (t <= -9e-27) {
		tmp = x;
	} else if (t <= -2.5e-306) {
		tmp = t_1;
	} else if (t <= 2.8e-252) {
		tmp = (z * -x) / t;
	} else if (t <= 4.1e-184) {
		tmp = t_1;
	} else if (t <= 1.15e-93) {
		tmp = z * (-x / t);
	} else if (t <= 1.45e+33) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / t) * y
    if (t <= (-9d-27)) then
        tmp = x
    else if (t <= (-2.5d-306)) then
        tmp = t_1
    else if (t <= 2.8d-252) then
        tmp = (z * -x) / t
    else if (t <= 4.1d-184) then
        tmp = t_1
    else if (t <= 1.15d-93) then
        tmp = z * (-x / t)
    else if (t <= 1.45d+33) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if (t <= -9e-27) {
		tmp = x;
	} else if (t <= -2.5e-306) {
		tmp = t_1;
	} else if (t <= 2.8e-252) {
		tmp = (z * -x) / t;
	} else if (t <= 4.1e-184) {
		tmp = t_1;
	} else if (t <= 1.15e-93) {
		tmp = z * (-x / t);
	} else if (t <= 1.45e+33) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * y
	tmp = 0
	if t <= -9e-27:
		tmp = x
	elif t <= -2.5e-306:
		tmp = t_1
	elif t <= 2.8e-252:
		tmp = (z * -x) / t
	elif t <= 4.1e-184:
		tmp = t_1
	elif t <= 1.15e-93:
		tmp = z * (-x / t)
	elif t <= 1.45e+33:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * y)
	tmp = 0.0
	if (t <= -9e-27)
		tmp = x;
	elseif (t <= -2.5e-306)
		tmp = t_1;
	elseif (t <= 2.8e-252)
		tmp = Float64(Float64(z * Float64(-x)) / t);
	elseif (t <= 4.1e-184)
		tmp = t_1;
	elseif (t <= 1.15e-93)
		tmp = Float64(z * Float64(Float64(-x) / t));
	elseif (t <= 1.45e+33)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * y;
	tmp = 0.0;
	if (t <= -9e-27)
		tmp = x;
	elseif (t <= -2.5e-306)
		tmp = t_1;
	elseif (t <= 2.8e-252)
		tmp = (z * -x) / t;
	elseif (t <= 4.1e-184)
		tmp = t_1;
	elseif (t <= 1.15e-93)
		tmp = z * (-x / t);
	elseif (t <= 1.45e+33)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t, -9e-27], x, If[LessEqual[t, -2.5e-306], t$95$1, If[LessEqual[t, 2.8e-252], N[(N[(z * (-x)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 4.1e-184], t$95$1, If[LessEqual[t, 1.15e-93], N[(z * N[((-x) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+33], t$95$1, x]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.0000000000000003e-27 or 1.45000000000000012e33 < t

   1. Initial program 83.9%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around inf 64.5%

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

   if -9.0000000000000003e-27 < t < -2.49999999999999999e-306 or 2.80000000000000018e-252 < t < 4.1e-184 or 1.1499999999999999e-93 < t < 1.45000000000000012e33

   1. Initial program 99.1%

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

      1. associate-*l/ 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in z around inf 66.2%

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

   5. Taylor expanded in y around inf 48.4%

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

      1. clear-num 48.4%

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

      2. un-div-inv 50.8%

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

   7. Applied egg-rr 50.8%

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

      1. associate-/r/ 60.8%

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

   9. Applied egg-rr 60.8%

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

   if -2.49999999999999999e-306 < t < 2.80000000000000018e-252

   1. Initial program 99.5%

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

      1. associate-*l/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in z around inf 75.9%

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

   5. Taylor expanded in y around 0 75.9%

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

      1. neg-mul-1 75.9%

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

      2. +-commutative 75.9%

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

      3. sub-neg 75.9%

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

      4. div-sub 84.2%

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

   7. Simplified 84.2%

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

      1. clear-num 84.1%

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

      2. div-inv 84.3%

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

   9. Applied egg-rr 84.3%

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

   10. Taylor expanded in y around 0 95.9%

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

       1. associate-*r/ 95.9%

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

       2. mul-1-neg 95.9%

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

       3. distribute-lft-neg-out 95.9%

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

   12. Simplified 95.9%

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

   if 4.1e-184 < t < 1.1499999999999999e-93

   1. Initial program 93.8%

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

      1. associate-*l/ 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in z around inf 76.2%

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

   5. Taylor expanded in y around 0 56.0%

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

      1. neg-mul-1 56.0%

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

      2. distribute-neg-frac 56.0%

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

   7. Simplified 56.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-306}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{t}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-93}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+33}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-225} \lor \neg \left(z \leq 4.5 \cdot 10^{-140}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.8e-225) (not (<= z 4.5e-140)))
   (+ x (* z (/ (- y x) t)))
   (+ x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.8e-225) || !(z <= 4.5e-140)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((z * y) / 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 <= (-7.8d-225)) .or. (.not. (z <= 4.5d-140))) then
        tmp = x + (z * ((y - x) / t))
    else
        tmp = x + ((z * y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.8e-225) || !(z <= 4.5e-140)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.8e-225) or not (z <= 4.5e-140):
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = x + ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.8e-225) || !(z <= 4.5e-140))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7.8e-225) || ~((z <= 4.5e-140)))
		tmp = x + (z * ((y - x) / t));
	else
		tmp = x + ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.8e-225], N[Not[LessEqual[z, 4.5e-140]], $MachinePrecision]], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8000000000000001e-225 or 4.50000000000000004e-140 < z

   1. Initial program 89.0%

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

      1. associate-*l/ 97.0%

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

   3. Simplified 97.0%

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

   if -7.8000000000000001e-225 < z < 4.50000000000000004e-140

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 96.7%

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

      1. *-commutative 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-225} \lor \neg \left(z \leq 4.5 \cdot 10^{-140}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]

Alternative 5: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-225}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-140}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.8e-225)
   (+ x (* z (/ (- y x) t)))
   (if (<= z 5.2e-140) (+ x (/ (* z y) t)) (+ x (/ z (/ t (- y x)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e-225) {
		tmp = x + (z * ((y - x) / t));
	} else if (z <= 5.2e-140) {
		tmp = x + ((z * y) / t);
	} else {
		tmp = x + (z / (t / (y - 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 <= (-7.8d-225)) then
        tmp = x + (z * ((y - x) / t))
    else if (z <= 5.2d-140) then
        tmp = x + ((z * y) / t)
    else
        tmp = x + (z / (t / (y - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.8e-225) {
		tmp = x + (z * ((y - x) / t));
	} else if (z <= 5.2e-140) {
		tmp = x + ((z * y) / t);
	} else {
		tmp = x + (z / (t / (y - x)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.8e-225:
		tmp = x + (z * ((y - x) / t))
	elif z <= 5.2e-140:
		tmp = x + ((z * y) / t)
	else:
		tmp = x + (z / (t / (y - x)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.8e-225)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	elseif (z <= 5.2e-140)
		tmp = Float64(x + Float64(Float64(z * y) / t));
	else
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.8e-225)
		tmp = x + (z * ((y - x) / t));
	elseif (z <= 5.2e-140)
		tmp = x + ((z * y) / t);
	else
		tmp = x + (z / (t / (y - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.8e-225], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-140], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.8000000000000001e-225

   1. Initial program 88.3%

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

      1. associate-*l/ 97.2%

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

   3. Simplified 97.2%

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

   if -7.8000000000000001e-225 < z < 5.1999999999999996e-140

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 96.7%

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

      1. *-commutative 96.7%

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

   4. Simplified 96.7%

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

   if 5.1999999999999996e-140 < z

   1. Initial program 89.7%

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

      1. associate-*l/ 96.8%

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

   3. Simplified 96.8%

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

      1. *-commutative 96.8%

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

      2. clear-num 96.7%

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

      3. un-div-inv 97.7%

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

   5. Applied egg-rr 97.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-225}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-140}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 6: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-24} \lor \neg \left(z \leq 7.5 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.1e-24) (not (<= z 7.5e-58))) (* z (/ (- y x) t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.1e-24) || !(z <= 7.5e-58)) {
		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 <= (-2.1d-24)) .or. (.not. (z <= 7.5d-58))) 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 <= -2.1e-24) || !(z <= 7.5e-58)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.1e-24) or not (z <= 7.5e-58):
		tmp = z * ((y - x) / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.1e-24) || !(z <= 7.5e-58))
		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 <= -2.1e-24) || ~((z <= 7.5e-58)))
		tmp = z * ((y - x) / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.1e-24], N[Not[LessEqual[z, 7.5e-58]], $MachinePrecision]], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.0999999999999999e-24 or 7.50000000000000002e-58 < z

   1. Initial program 85.3%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 81.3%

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

   5. Taylor expanded in y around 0 81.3%

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

      1. neg-mul-1 81.3%

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

      2. +-commutative 81.3%

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

      3. sub-neg 81.3%

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

      4. div-sub 83.5%

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

   7. Simplified 83.5%

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

   if -2.0999999999999999e-24 < z < 7.50000000000000002e-58

   1. Initial program 98.4%

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

      1. associate-*l/ 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in t around inf 62.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-24} \lor \neg \left(z \leq 7.5 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -110000 \lor \neg \left(z \leq 9.2 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -110000.0) (not (<= z 9.2e-58)))
   (* z (/ (- y x) t))
   (+ x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -110000.0) || !(z <= 9.2e-58)) {
		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 <= (-110000.0d0)) .or. (.not. (z <= 9.2d-58))) 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 <= -110000.0) || !(z <= 9.2e-58)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x + ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -110000.0) or not (z <= 9.2e-58):
		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 <= -110000.0) || !(z <= 9.2e-58))
		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 <= -110000.0) || ~((z <= 9.2e-58)))
		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, -110000.0], N[Not[LessEqual[z, 9.2e-58]], $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 < -1.1e5 or 9.1999999999999995e-58 < z

   1. Initial program 84.3%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 82.4%

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

   5. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. +-commutative 82.4%

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

      3. sub-neg 82.4%

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

      4. div-sub 84.7%

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

   7. Simplified 84.7%

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

   if -1.1e5 < z < 9.1999999999999995e-58

   1. Initial program 98.5%

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

      1. associate-*l/ 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in y around inf 86.0%

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

      1. associate-*r/ 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -110000 \lor \neg \left(z \leq 9.2 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \end{array} \]

Alternative 8: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -85000 \lor \neg \left(z \leq 8.8 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -85000.0) (not (<= z 8.8e-58)))
   (* z (/ (- y x) t))
   (+ x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -85000.0) || !(z <= 8.8e-58)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x + ((z * y) / 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 <= (-85000.0d0)) .or. (.not. (z <= 8.8d-58))) then
        tmp = z * ((y - x) / t)
    else
        tmp = x + ((z * y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -85000.0) || !(z <= 8.8e-58)) {
		tmp = z * ((y - x) / t);
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -85000.0) or not (z <= 8.8e-58):
		tmp = z * ((y - x) / t)
	else:
		tmp = x + ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -85000.0) || !(z <= 8.8e-58))
		tmp = Float64(z * Float64(Float64(y - x) / t));
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -85000.0) || ~((z <= 8.8e-58)))
		tmp = z * ((y - x) / t);
	else
		tmp = x + ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -85000.0], N[Not[LessEqual[z, 8.8e-58]], $MachinePrecision]], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -85000 or 8.80000000000000023e-58 < z

   1. Initial program 84.3%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 82.4%

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

   5. Taylor expanded in y around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. +-commutative 82.4%

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

      3. sub-neg 82.4%

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

      4. div-sub 84.7%

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

   7. Simplified 84.7%

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

   if -85000 < z < 8.80000000000000023e-58

   1. Initial program 98.5%

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

   2. Taylor expanded in y around inf 86.0%

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

      1. *-commutative 86.0%

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

   4. Simplified 86.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -85000 \lor \neg \left(z \leq 8.8 \cdot 10^{-58}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]

Alternative 9: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-57} \lor \neg \left(y \leq 5.5 \cdot 10^{+67}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.3e-57) (not (<= y 5.5e+67)))
   (+ x (* (/ z t) y))
   (- x (* (/ z t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.3e-57) || !(y <= 5.5e+67)) {
		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 <= (-3.3d-57)) .or. (.not. (y <= 5.5d+67))) 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 <= -3.3e-57) || !(y <= 5.5e+67)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x - ((z / t) * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -3.3e-57) or not (y <= 5.5e+67):
		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 <= -3.3e-57) || !(y <= 5.5e+67))
		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 <= -3.3e-57) || ~((y <= 5.5e+67)))
		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, -3.3e-57], N[Not[LessEqual[y, 5.5e+67]], $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 < -3.2999999999999998e-57 or 5.49999999999999968e67 < y

   1. Initial program 89.1%

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

      1. associate-*l/ 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in y around inf 85.3%

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

      1. associate-*r/ 89.6%

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

   6. Simplified 89.6%

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

   if -3.2999999999999998e-57 < y < 5.49999999999999968e67

   1. Initial program 94.0%

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

      1. associate-*l/ 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. distribute-lft-in 88.2%

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

      2. mul-1-neg 88.2%

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

      3. distribute-rgt-neg-in 88.2%

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

      4. unsub-neg 88.2%

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

      5. *-rgt-identity 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-57} \lor \neg \left(y \leq 5.5 \cdot 10^{+67}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot x\\ \end{array} \]

Alternative 10: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-20} \lor \neg \left(z \leq 1.08 \cdot 10^{-54}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e-20) (not (<= z 1.08e-54))) (* z (/ y t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e-20) || !(z <= 1.08e-54)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.5d-20)) .or. (.not. (z <= 1.08d-54))) then
        tmp = z * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e-20) || !(z <= 1.08e-54)) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e-20) or not (z <= 1.08e-54):
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e-20) || !(z <= 1.08e-54))
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.5e-20) || ~((z <= 1.08e-54)))
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e-20], N[Not[LessEqual[z, 1.08e-54]], $MachinePrecision]], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999996e-20 or 1.08000000000000002e-54 < z

   1. Initial program 85.2%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 81.1%

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

   5. Taylor expanded in y around inf 52.6%

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

   if -5.4999999999999996e-20 < z < 1.08000000000000002e-54

   1. Initial program 98.4%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in t around inf 62.2%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-20} \lor \neg \left(z \leq 1.08 \cdot 10^{-54}\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 55.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-23} \lor \neg \left(z \leq 1.02 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.6e-23) (not (<= z 1.02e-51))) (* (/ z t) y) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e-23) || !(z <= 1.02e-51)) {
		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 ((z <= (-2.6d-23)) .or. (.not. (z <= 1.02d-51))) 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 ((z <= -2.6e-23) || !(z <= 1.02e-51)) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.6e-23) or not (z <= 1.02e-51):
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.6e-23) || !(z <= 1.02e-51))
		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 ((z <= -2.6e-23) || ~((z <= 1.02e-51)))
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.6e-23], N[Not[LessEqual[z, 1.02e-51]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e-23 or 1.01999999999999998e-51 < z

   1. Initial program 85.2%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 81.1%

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

   5. Taylor expanded in y around inf 52.6%

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

      1. clear-num 52.5%

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

      2. un-div-inv 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-/r/ 53.4%

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

   9. Applied egg-rr 53.4%

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

   if -2.6e-23 < z < 1.01999999999999998e-51

   1. Initial program 98.4%

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

      1. associate-*l/ 80.9%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in t around inf 62.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-23} \lor \neg \left(z \leq 1.02 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 97.4% 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 91.5%

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

   1. associate-/l* 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 13: 38.8% 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 91.5%

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

   1. associate-*l/ 90.7%

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

3. Simplified 90.7%

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

4. Taylor expanded in t around inf 39.4%

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

5. Final simplification 39.4%

   \[\leadsto x \]

Developer target: 97.7% 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 2023290 
(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)))