Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.6% → 98.8%

Time: 9.1s

Alternatives: 11

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: 92.6% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+297} \lor \neg \left(t_1 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))))
   (if (or (<= t_1 -5e+297) (not (<= t_1 5e+307)))
     (+ x (/ z (/ t (- y x))))
     t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double tmp;
	if ((t_1 <= -5e+297) || !(t_1 <= 5e+307)) {
		tmp = x + (z / (t / (y - x)));
	} else {
		tmp = t_1;
	}
	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 = x + (((y - x) * z) / t)
    if ((t_1 <= (-5d+297)) .or. (.not. (t_1 <= 5d+307))) then
        tmp = x + (z / (t / (y - x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double tmp;
	if ((t_1 <= -5e+297) || !(t_1 <= 5e+307)) {
		tmp = x + (z / (t / (y - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (((y - x) * z) / t)
	tmp = 0
	if (t_1 <= -5e+297) or not (t_1 <= 5e+307):
		tmp = x + (z / (t / (y - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	tmp = 0.0
	if ((t_1 <= -5e+297) || !(t_1 <= 5e+307))
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (((y - x) * z) / t);
	tmp = 0.0;
	if ((t_1 <= -5e+297) || ~((t_1 <= 5e+307)))
		tmp = x + (z / (t / (y - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+297], N[Not[LessEqual[t$95$1, 5e+307]], $MachinePrecision]], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -4.9999999999999998e297 or 5e307 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 76.4%

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

      1. associate-/l* 99.8%

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

      2. clear-num 99.8%

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

      3. inv-pow 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. clear-num 99.8%

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

      3. associate-/r/ 99.9%

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

      4. clear-num 99.9%

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

      5. associate-*l/ 99.9%

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

      6. *-un-lft-identity 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -4.9999999999999998e297 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 5e307

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -5 \cdot 10^{+297} \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-99} \lor \neg \left(y \leq 1.2 \cdot 10^{-108}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.5e-99) (not (<= y 1.2e-108)))
   (+ x (* y (/ z t)))
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.5e-99) || !(y <= 1.2e-108)) {
		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 <= (-1.5d-99)) .or. (.not. (y <= 1.2d-108))) 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 <= -1.5e-99) || !(y <= 1.2e-108)) {
		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 <= -1.5e-99) or not (y <= 1.2e-108):
		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 <= -1.5e-99) || !(y <= 1.2e-108))
		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 <= -1.5e-99) || ~((y <= 1.2e-108)))
		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, -1.5e-99], N[Not[LessEqual[y, 1.2e-108]], $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 < -1.50000000000000003e-99 or 1.20000000000000009e-108 < y

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf 83.7%

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

      1. associate-*r/ 88.7%

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

   5. Simplified 88.7%

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

   if -1.50000000000000003e-99 < y < 1.20000000000000009e-108

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf 90.4%

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

      1. mul-1-neg 90.4%

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

      2. unsub-neg 90.4%

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

   5. Simplified 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-99} \lor \neg \left(y \leq 1.2 \cdot 10^{-108}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-99} \lor \neg \left(y \leq 6 \cdot 10^{-109}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.2e-99) (not (<= y 6e-109)))
   (+ x (* y (/ z t)))
   (- x (/ x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.2e-99) || !(y <= 6e-109)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (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 ((y <= (-5.2d-99)) .or. (.not. (y <= 6d-109))) then
        tmp = x + (y * (z / t))
    else
        tmp = x - (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 ((y <= -5.2e-99) || !(y <= 6e-109)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (x / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.2e-99) or not (y <= 6e-109):
		tmp = x + (y * (z / t))
	else:
		tmp = x - (x / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.2e-99) || !(y <= 6e-109))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x - Float64(x / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -5.2e-99) || ~((y <= 6e-109)))
		tmp = x + (y * (z / t));
	else
		tmp = x - (x / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.2e-99], N[Not[LessEqual[y, 6e-109]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.2000000000000001e-99 or 6.00000000000000043e-109 < y

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf 83.7%

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

      1. associate-*r/ 88.7%

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

   5. Simplified 88.7%

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

   if -5.2000000000000001e-99 < y < 6.00000000000000043e-109

   1. Initial program 94.7%

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

      1. associate-*l/ 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in y around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. distribute-neg-frac 92.5%

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

   7. Simplified 92.5%

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

      1. add-sqr-sqrt 44.0%

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

      2. sqrt-unprod 54.0%

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

      3. sqr-neg 54.0%

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

      4. sqrt-unprod 26.1%

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

      5. add-sqr-sqrt 52.0%

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

      6. cancel-sign-sub 52.0%

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

      7. distribute-frac-neg 52.0%

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

      8. associate-*l/ 47.6%

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

      9. associate-/l* 53.0%

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

      10. add-sqr-sqrt 25.9%

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

      11. sqrt-unprod 56.9%

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

      12. sqr-neg 56.9%

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

      13. sqrt-unprod 47.4%

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

      14. add-sqr-sqrt 90.4%

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

   9. Applied egg-rr 90.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-99} \lor \neg \left(y \leq 6 \cdot 10^{-109}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-99} \lor \neg \left(y \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.2e-99) (not (<= y 2.9e-108)))
   (+ x (* y (/ z t)))
   (- x (* z (/ x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.2e-99) || !(y <= 2.9e-108)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (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 ((y <= (-3.2d-99)) .or. (.not. (y <= 2.9d-108))) then
        tmp = x + (y * (z / t))
    else
        tmp = x - (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 ((y <= -3.2e-99) || !(y <= 2.9e-108)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = x - (z * (x / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -3.2e-99) or not (y <= 2.9e-108):
		tmp = x + (y * (z / t))
	else:
		tmp = x - (z * (x / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.2e-99) || !(y <= 2.9e-108))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(x - Float64(z * Float64(x / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -3.2e-99) || ~((y <= 2.9e-108)))
		tmp = x + (y * (z / t));
	else
		tmp = x - (z * (x / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.2e-99], N[Not[LessEqual[y, 2.9e-108]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2000000000000001e-99 or 2.9000000000000001e-108 < y

   1. Initial program 91.4%

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

   3. Taylor expanded in y around inf 83.7%

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

      1. associate-*r/ 88.7%

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

   5. Simplified 88.7%

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

   if -3.2000000000000001e-99 < y < 2.9000000000000001e-108

   1. Initial program 94.7%

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

      1. associate-*l/ 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in y around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. distribute-neg-frac 92.5%

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

   7. Simplified 92.5%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-99} \lor \neg \left(y \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+80}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 8.5e+80) (+ x (* z (/ (- y x) t))) (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.5e+80) {
		tmp = x + (z * ((y - x) / 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 (y <= 8.5d+80) then
        tmp = x + (z * ((y - x) / 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 (y <= 8.5e+80) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8.5e+80:
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8.5e+80)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / 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 (y <= 8.5e+80)
		tmp = x + (z * ((y - x) / t));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8.5e+80], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000007e80

   1. Initial program 93.0%

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

      1. associate-*l/ 94.9%

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

   4. Applied egg-rr 94.9%

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

   if 8.50000000000000007e80 < y

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf 90.0%

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

      1. associate-*r/ 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+80}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.26e+88) (+ x (/ z (/ t (- y x)))) (+ x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.26e+88) {
		tmp = x + (z / (t / (y - x)));
	} 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 (y <= 1.26d+88) then
        tmp = x + (z / (t / (y - x)))
    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 (y <= 1.26e+88) {
		tmp = x + (z / (t / (y - x)));
	} else {
		tmp = x + (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.26e+88:
		tmp = x + (z / (t / (y - x)))
	else:
		tmp = x + (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.26e+88)
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	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 (y <= 1.26e+88)
		tmp = x + (z / (t / (y - x)));
	else
		tmp = x + (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.26e+88], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.26e88

   1. Initial program 93.0%

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

      1. associate-/l* 96.9%

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

      2. clear-num 96.9%

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

      3. inv-pow 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. unpow-1 96.9%

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

      2. clear-num 96.9%

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

      3. associate-/r/ 94.9%

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

      4. clear-num 94.6%

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

      5. associate-*l/ 95.4%

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

      6. *-un-lft-identity 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 1.26e88 < y

   1. Initial program 90.0%

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

   3. Taylor expanded in y around inf 90.0%

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

      1. associate-*r/ 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{+88}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-51}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.5e-18) x (if (<= t 7.8e-51) (* z (/ (- x) t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.5e-18) {
		tmp = x;
	} else if (t <= 7.8e-51) {
		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 (t <= (-6.5d-18)) then
        tmp = x
    else if (t <= 7.8d-51) 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 (t <= -6.5e-18) {
		tmp = x;
	} else if (t <= 7.8e-51) {
		tmp = z * (-x / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.5e-18:
		tmp = x
	elif t <= 7.8e-51:
		tmp = z * (-x / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.5e-18)
		tmp = x;
	elseif (t <= 7.8e-51)
		tmp = Float64(z * Float64(Float64(-x) / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.5e-18)
		tmp = x;
	elseif (t <= 7.8e-51)
		tmp = z * (-x / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.5e-18], x, If[LessEqual[t, 7.8e-51], N[(z * N[((-x) / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -6.50000000000000008e-18 or 7.7999999999999995e-51 < t

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0 61.4%

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

   if -6.50000000000000008e-18 < t < 7.7999999999999995e-51

   1. Initial program 99.9%

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

      1. associate-*l/ 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in y around 0 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   7. Simplified 54.5%

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

      1. add-sqr-sqrt 23.8%

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

      2. sqrt-unprod 27.1%

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

      3. sqr-neg 27.1%

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

      4. sqrt-unprod 7.3%

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

      5. add-sqr-sqrt 14.1%

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

      6. cancel-sign-sub 14.1%

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

      7. distribute-frac-neg 14.1%

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

      8. associate-*l/ 18.2%

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

      9. associate-/l* 20.6%

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

      10. add-sqr-sqrt 10.9%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 31.3%

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

      14. add-sqr-sqrt 59.3%

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

   9. Applied egg-rr 59.3%

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

   10. Taylor expanded in t around 0 45.8%

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

       1. mul-1-neg 45.8%

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

       2. associate-*l/ 45.5%

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

       3. *-commutative 45.5%

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

       4. distribute-rgt-neg-in 45.5%

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

   12. Simplified 45.5%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-51}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(-\frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7e-18) x (if (<= t 1.25e-33) (* x (- (/ z t))) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7e-18) {
		tmp = x;
	} else if (t <= 1.25e-33) {
		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 (t <= (-7d-18)) then
        tmp = x
    else if (t <= 1.25d-33) 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 (t <= -7e-18) {
		tmp = x;
	} else if (t <= 1.25e-33) {
		tmp = x * -(z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7e-18:
		tmp = x
	elif t <= 1.25e-33:
		tmp = x * -(z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7e-18)
		tmp = x;
	elseif (t <= 1.25e-33)
		tmp = Float64(x * Float64(-Float64(z / t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7e-18)
		tmp = x;
	elseif (t <= 1.25e-33)
		tmp = x * -(z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7e-18], x, If[LessEqual[t, 1.25e-33], N[(x * (-N[(z / t), $MachinePrecision])), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -6.9999999999999997e-18 or 1.25000000000000007e-33 < t

   1. Initial program 86.4%

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

   3. Taylor expanded in z around 0 61.4%

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

   if -6.9999999999999997e-18 < t < 1.25000000000000007e-33

   1. Initial program 99.9%

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

      1. associate-*l/ 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in y around 0 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   7. Simplified 54.5%

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

      1. add-sqr-sqrt 23.8%

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

      2. sqrt-unprod 27.1%

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

      3. sqr-neg 27.1%

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

      4. sqrt-unprod 7.3%

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

      5. add-sqr-sqrt 14.1%

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

      6. cancel-sign-sub 14.1%

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

      7. distribute-frac-neg 14.1%

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

      8. associate-*l/ 18.2%

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

      9. associate-/l* 20.6%

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

      10. add-sqr-sqrt 10.9%

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

      11. sqrt-unprod 31.4%

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

      12. sqr-neg 31.4%

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

      13. sqrt-unprod 31.3%

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

      14. add-sqr-sqrt 59.3%

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

   9. Applied egg-rr 59.3%

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

   10. Taylor expanded in t around 0 45.8%

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

       1. mul-1-neg 45.8%

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

       2. associate-*r/ 45.9%

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

       3. distribute-lft-neg-out 45.9%

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

       4. *-commutative 45.9%

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

   12. Simplified 45.9%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(-\frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% 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 92.6%

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

   1. associate-/l* 97.4%

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

3. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 10: 65.7% 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 92.6%

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

3. Taylor expanded in x around inf 64.6%

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

   1. mul-1-neg 64.6%

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

   2. unsub-neg 64.6%

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

5. Simplified 64.6%

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

6. Final simplification 64.6%

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

Alternative 11: 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 92.6%

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

3. Taylor expanded in z around 0 40.5%

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

4. Final simplification 40.5%

   \[\leadsto x \]
5. Add Preprocessing

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 2024013 
(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)))