Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 7

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 60.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -53000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -6.5e+129)
     (* y z)
     (if (<= z -1.65e+98)
       t_0
       (if (<= z -53000.0)
         (* y z)
         (if (<= z 6e-68) x (if (<= z 1.15e+56) (* y z) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -6.5e+129) {
		tmp = y * z;
	} else if (z <= -1.65e+98) {
		tmp = t_0;
	} else if (z <= -53000.0) {
		tmp = y * z;
	} else if (z <= 6e-68) {
		tmp = x;
	} else if (z <= 1.15e+56) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (z <= (-6.5d+129)) then
        tmp = y * z
    else if (z <= (-1.65d+98)) then
        tmp = t_0
    else if (z <= (-53000.0d0)) then
        tmp = y * z
    else if (z <= 6d-68) then
        tmp = x
    else if (z <= 1.15d+56) then
        tmp = y * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -6.5e+129) {
		tmp = y * z;
	} else if (z <= -1.65e+98) {
		tmp = t_0;
	} else if (z <= -53000.0) {
		tmp = y * z;
	} else if (z <= 6e-68) {
		tmp = x;
	} else if (z <= 1.15e+56) {
		tmp = y * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -6.5e+129:
		tmp = y * z
	elif z <= -1.65e+98:
		tmp = t_0
	elif z <= -53000.0:
		tmp = y * z
	elif z <= 6e-68:
		tmp = x
	elif z <= 1.15e+56:
		tmp = y * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -6.5e+129)
		tmp = Float64(y * z);
	elseif (z <= -1.65e+98)
		tmp = t_0;
	elseif (z <= -53000.0)
		tmp = Float64(y * z);
	elseif (z <= 6e-68)
		tmp = x;
	elseif (z <= 1.15e+56)
		tmp = Float64(y * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -6.5e+129)
		tmp = y * z;
	elseif (z <= -1.65e+98)
		tmp = t_0;
	elseif (z <= -53000.0)
		tmp = y * z;
	elseif (z <= 6e-68)
		tmp = x;
	elseif (z <= 1.15e+56)
		tmp = y * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -6.5e+129], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.65e+98], t$95$0, If[LessEqual[z, -53000.0], N[(y * z), $MachinePrecision], If[LessEqual[z, 6e-68], x, If[LessEqual[z, 1.15e+56], N[(y * z), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.4999999999999995e129 or -1.65000000000000014e98 < z < -53000 or 6e-68 < z < 1.15000000000000007e56

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.8%

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

   if -6.4999999999999995e129 < z < -1.65000000000000014e98 or 1.15000000000000007e56 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. unsub-neg 71.5%

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

   5. Simplified 71.5%

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

   6. Taylor expanded in z around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. distribute-rgt-neg-out 71.5%

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

   8. Simplified 71.5%

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

   if -53000 < z < 6e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 77.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+129}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -53000:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-100} \lor \neg \left(x \leq 7.4 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e-100) (not (<= x 7.4e-116))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-100) || !(x <= 7.4e-116)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2d-100)) .or. (.not. (x <= 7.4d-116))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-100) || !(x <= 7.4e-116)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e-100) or not (x <= 7.4e-116):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e-100) || !(x <= 7.4e-116))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e-100) || ~((x <= 7.4e-116)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e-100], N[Not[LessEqual[x, 7.4e-116]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-100 or 7.4000000000000005e-116 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.6%

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

      1. mul-1-neg 83.6%

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

      2. unsub-neg 83.6%

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

   5. Simplified 83.6%

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

   if -2e-100 < x < 7.4000000000000005e-116

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 77.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-100} \lor \neg \left(x \leq 7.4 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -50000 \lor \neg \left(z \leq 6.4 \cdot 10^{-68}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -50000.0) (not (<= z 6.4e-68))) (* (- y x) z) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -50000.0) || !(z <= 6.4e-68)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-50000.0d0)) .or. (.not. (z <= 6.4d-68))) then
        tmp = (y - x) * z
    else
        tmp = x * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -50000.0) || !(z <= 6.4e-68)) {
		tmp = (y - x) * z;
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -50000.0) or not (z <= 6.4e-68):
		tmp = (y - x) * z
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -50000.0) || !(z <= 6.4e-68))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(x * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -50000.0) || ~((z <= 6.4e-68)))
		tmp = (y - x) * z;
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -50000.0], N[Not[LessEqual[z, 6.4e-68]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5e4 or 6.3999999999999998e-68 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 96.9%

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

   if -5e4 < z < 6.3999999999999998e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

      1. mul-1-neg 79.1%

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

      2. unsub-neg 79.1%

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

   5. Simplified 79.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -50000 \lor \neg \left(z \leq 6.4 \cdot 10^{-68}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -47000 \lor \neg \left(z \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -47000.0) (not (<= z 4.9e-68))) (* y z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -47000.0) || !(z <= 4.9e-68)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-47000.0d0)) .or. (.not. (z <= 4.9d-68))) then
        tmp = y * z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -47000.0) || !(z <= 4.9e-68)) {
		tmp = y * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -47000.0) or not (z <= 4.9e-68):
		tmp = y * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -47000.0) || !(z <= 4.9e-68))
		tmp = Float64(y * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -47000.0) || ~((z <= 4.9e-68)))
		tmp = y * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -47000.0], N[Not[LessEqual[z, 4.9e-68]], $MachinePrecision]], N[(y * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -47000 or 4.89999999999999977e-68 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 51.9%

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

   if -47000 < z < 4.89999999999999977e-68

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 77.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -47000 \lor \neg \left(z \leq 4.9 \cdot 10^{-68}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (- y x) z)))

C

double code(double x, double y, double z) {
	return x + ((y - x) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y - x) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto x + \left(y - x\right) \cdot z \]
4. Add Preprocessing

Alternative 7: 36.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around 0 39.6%

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

4. Final simplification 39.6%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))