Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 84.1% → 97.0%

Time: 6.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-/l* 97.4%

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

3. Simplified 97.4%

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

   1. clear-num 97.1%

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

   2. un-div-inv 97.4%

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 2: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-63} \lor \neg \left(z \leq 2.35 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.8e-63) (not (<= z 2.35e-38)))
   (* x (- 1.0 (/ y z)))
   (/ x (/ t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e-63) || !(z <= 2.35e-38)) {
		tmp = x * (1.0 - (y / z));
	} else {
		tmp = x / (t / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.8d-63)) .or. (.not. (z <= 2.35d-38))) then
        tmp = x * (1.0d0 - (y / z))
    else
        tmp = x / (t / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.8e-63) || !(z <= 2.35e-38)) {
		tmp = x * (1.0 - (y / z));
	} else {
		tmp = x / (t / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.8e-63) or not (z <= 2.35e-38):
		tmp = x * (1.0 - (y / z))
	else:
		tmp = x / (t / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.8e-63) || !(z <= 2.35e-38))
		tmp = Float64(x * Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(x / Float64(t / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.8e-63) || ~((z <= 2.35e-38)))
		tmp = x * (1.0 - (y / z));
	else
		tmp = x / (t / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.8e-63], N[Not[LessEqual[z, 2.35e-38]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000002e-63 or 2.34999999999999999e-38 < z

   1. Initial program 78.7%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. associate-/l* 72.9%

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

      3. distribute-rgt-neg-in 72.9%

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

      4. distribute-frac-neg 72.9%

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

      5. neg-sub0 72.9%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 72.9%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 72.9%

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

      8. mul-1-neg 72.9%

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

      9. +-commutative 72.9%

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

      10. mul-1-neg 72.9%

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

      11. sub-neg 72.9%

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

      12. div-sub 72.9%

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

      13. *-inverses 72.9%

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

   7. Simplified 72.9%

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

   if -2.8000000000000002e-63 < z < 2.34999999999999999e-38

   1. Initial program 90.3%

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

      1. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

      1. clear-num 93.0%

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

      2. un-div-inv 93.6%

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

   6. Applied egg-rr 93.6%

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

   7. Taylor expanded in z around 0 74.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-63} \lor \neg \left(z \leq 2.35 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-18} \lor \neg \left(z \leq 0.0041\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.4e-18) (not (<= z 0.0041)))
   (* x (- 1.0 (/ y z)))
   (* x (/ y (- t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.4e-18) || !(z <= 0.0041)) {
		tmp = x * (1.0 - (y / z));
	} else {
		tmp = x * (y / (t - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-7.4d-18)) .or. (.not. (z <= 0.0041d0))) then
        tmp = x * (1.0d0 - (y / z))
    else
        tmp = x * (y / (t - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.4e-18) || !(z <= 0.0041)) {
		tmp = x * (1.0 - (y / z));
	} else {
		tmp = x * (y / (t - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.4e-18) or not (z <= 0.0041):
		tmp = x * (1.0 - (y / z))
	else:
		tmp = x * (y / (t - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.4e-18) || !(z <= 0.0041))
		tmp = Float64(x * Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(x * Float64(y / Float64(t - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7.4e-18) || ~((z <= 0.0041)))
		tmp = x * (1.0 - (y / z));
	else
		tmp = x * (y / (t - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.4e-18], N[Not[LessEqual[z, 0.0041]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4000000000000007e-18 or 0.00410000000000000035 < z

   1. Initial program 77.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 59.8%

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

      1. mul-1-neg 59.8%

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

      2. associate-/l* 77.3%

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

      3. distribute-rgt-neg-in 77.3%

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

      4. distribute-frac-neg 77.3%

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

      5. neg-sub0 77.3%

         \[\leadsto x \cdot \frac{\color{blue}{0 - \left(y - z\right)}}{z} \]

      6. associate--r- 77.3%

         \[\leadsto x \cdot \frac{\color{blue}{\left(0 - y\right) + z}}{z} \]

      7. neg-sub0 77.3%

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

      8. mul-1-neg 77.3%

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

      9. +-commutative 77.3%

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

      10. mul-1-neg 77.3%

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

      11. sub-neg 77.3%

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

      12. div-sub 77.3%

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

      13. *-inverses 77.3%

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

   7. Simplified 77.3%

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

   if -7.4000000000000007e-18 < z < 0.00410000000000000035

   1. Initial program 89.6%

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

      1. associate-/l* 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 74.4%

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

      1. associate-/l* 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-18} \lor \neg \left(z \leq 0.0041\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-5} \lor \neg \left(y \leq 4.3 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.95e-5) (not (<= y 4.3e-36)))
   (* x (/ y (- t z)))
   (* x (/ z (- z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.95e-5) || !(y <= 4.3e-36)) {
		tmp = x * (y / (t - z));
	} else {
		tmp = x * (z / (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 <= (-1.95d-5)) .or. (.not. (y <= 4.3d-36))) then
        tmp = x * (y / (t - z))
    else
        tmp = x * (z / (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 <= -1.95e-5) || !(y <= 4.3e-36)) {
		tmp = x * (y / (t - z));
	} else {
		tmp = x * (z / (z - t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.95e-5) or not (y <= 4.3e-36):
		tmp = x * (y / (t - z))
	else:
		tmp = x * (z / (z - t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.95e-5) || !(y <= 4.3e-36))
		tmp = Float64(x * Float64(y / Float64(t - z)));
	else
		tmp = Float64(x * Float64(z / Float64(z - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.95e-5) || ~((y <= 4.3e-36)))
		tmp = x * (y / (t - z));
	else
		tmp = x * (z / (z - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.95e-5], N[Not[LessEqual[y, 4.3e-36]], $MachinePrecision]], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.95e-5 or 4.3000000000000002e-36 < y

   1. Initial program 83.9%

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

      1. associate-/l* 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in y around inf 69.0%

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

      1. associate-/l* 77.9%

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

   7. Simplified 77.9%

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

   if -1.95e-5 < y < 4.3000000000000002e-36

   1. Initial program 82.5%

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

      1. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. distribute-neg-frac2 67.5%

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

      3. neg-sub0 67.5%

         \[\leadsto \frac{x \cdot z}{\color{blue}{0 - \left(t - z\right)}} \]

      4. associate--r- 67.5%

         \[\leadsto \frac{x \cdot z}{\color{blue}{\left(0 - t\right) + z}} \]

      5. neg-sub0 67.5%

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

      6. +-commutative 67.5%

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

      7. sub-neg 67.5%

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

      8. associate-/l* 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-5} \lor \neg \left(y \leq 4.3 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5e-6)
   (/ x (/ (- t z) y))
   (if (<= y 1.35e-18) (* x (/ z (- z t))) (* x (/ y (- t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e-6) {
		tmp = x / ((t - z) / y);
	} else if (y <= 1.35e-18) {
		tmp = x * (z / (z - t));
	} else {
		tmp = x * (y / (t - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5d-6)) then
        tmp = x / ((t - z) / y)
    else if (y <= 1.35d-18) then
        tmp = x * (z / (z - t))
    else
        tmp = x * (y / (t - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5e-6) {
		tmp = x / ((t - z) / y);
	} else if (y <= 1.35e-18) {
		tmp = x * (z / (z - t));
	} else {
		tmp = x * (y / (t - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5e-6:
		tmp = x / ((t - z) / y)
	elif y <= 1.35e-18:
		tmp = x * (z / (z - t))
	else:
		tmp = x * (y / (t - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5e-6)
		tmp = Float64(x / Float64(Float64(t - z) / y));
	elseif (y <= 1.35e-18)
		tmp = Float64(x * Float64(z / Float64(z - t)));
	else
		tmp = Float64(x * Float64(y / Float64(t - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5e-6)
		tmp = x / ((t - z) / y);
	elseif (y <= 1.35e-18)
		tmp = x * (z / (z - t));
	else
		tmp = x * (y / (t - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5e-6], N[(x / N[(N[(t - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e-18], N[(x * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.00000000000000041e-6

   1. Initial program 83.9%

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

      1. associate-/l* 96.8%

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

   3. Simplified 96.8%

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

      1. clear-num 96.8%

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

      2. un-div-inv 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in y around inf 75.0%

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

   if -5.00000000000000041e-6 < y < 1.34999999999999994e-18

   1. Initial program 82.5%

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

      1. associate-/l* 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. distribute-neg-frac2 67.5%

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

      3. neg-sub0 67.5%

         \[\leadsto \frac{x \cdot z}{\color{blue}{0 - \left(t - z\right)}} \]

      4. associate--r- 67.5%

         \[\leadsto \frac{x \cdot z}{\color{blue}{\left(0 - t\right) + z}} \]

      5. neg-sub0 67.5%

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

      6. +-commutative 67.5%

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

      7. sub-neg 67.5%

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

      8. associate-/l* 83.1%

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

   7. Simplified 83.1%

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

   if 1.34999999999999994e-18 < y

   1. Initial program 83.9%

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

      1. associate-/l* 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in y around inf 69.5%

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

      1. associate-/l* 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 780:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.5e+92) x (if (<= z 780.0) (* x (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+92) {
		tmp = x;
	} else if (z <= 780.0) {
		tmp = x * (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 <= (-8.5d+92)) then
        tmp = x
    else if (z <= 780.0d0) then
        tmp = x * (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 <= -8.5e+92) {
		tmp = x;
	} else if (z <= 780.0) {
		tmp = x * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.5e+92:
		tmp = x
	elif z <= 780.0:
		tmp = x * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.5e+92)
		tmp = x;
	elseif (z <= 780.0)
		tmp = Float64(x * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -8.5e+92)
		tmp = x;
	elseif (z <= 780.0)
		tmp = x * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.5e+92], x, If[LessEqual[z, 780.0], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000001e92 or 780 < z

   1. Initial program 73.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 64.6%

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

   if -8.5000000000000001e92 < z < 780

   1. Initial program 90.3%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in z around 0 56.8%

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

      1. associate-/l* 62.0%

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

   7. Simplified 62.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 780:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+91}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.082:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.5e+91) x (if (<= z 0.082) (/ x (/ t y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.5e+91) {
		tmp = x;
	} else if (z <= 0.082) {
		tmp = x / (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 <= (-7.5d+91)) then
        tmp = x
    else if (z <= 0.082d0) then
        tmp = x / (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 <= -7.5e+91) {
		tmp = x;
	} else if (z <= 0.082) {
		tmp = x / (t / y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.5e+91:
		tmp = x
	elif z <= 0.082:
		tmp = x / (t / y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.5e+91)
		tmp = x;
	elseif (z <= 0.082)
		tmp = Float64(x / Float64(t / y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.5e+91)
		tmp = x;
	elseif (z <= 0.082)
		tmp = x / (t / y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.5e+91], x, If[LessEqual[z, 0.082], N[(x / N[(t / y), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000033e91 or 0.0820000000000000034 < z

   1. Initial program 73.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 64.6%

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

   if -7.50000000000000033e91 < z < 0.0820000000000000034

   1. Initial program 90.3%

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

      1. associate-/l* 95.5%

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

   3. Simplified 95.5%

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

      1. clear-num 95.2%

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

      2. un-div-inv 95.6%

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

   6. Applied egg-rr 95.6%

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

   7. Taylor expanded in z around 0 62.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+91}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.082:\\ \;\;\;\;\frac{x}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y - z}{t - z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. associate-/l* 97.4%

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

3. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 9: 34.9% 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 83.3%

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

   1. associate-/l* 97.4%

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

3. Simplified 97.4%

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

5. Taylor expanded in z around inf 34.7%

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

6. Final simplification 34.7%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{t - z}{y - z}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :alt
  (/ x (/ (- t z) (- y z)))

  (/ (* x (- y z)) (- t z)))