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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 20.9% 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-define 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: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+77} \lor \neg \left(x \leq 4.4 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.2e+77) (not (<= x 4.4e+98))) (* x (- 1.0 z)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.2e+77) || !(x <= 4.4e+98)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = x + (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 <= (-6.2d+77)) .or. (.not. (x <= 4.4d+98))) then
        tmp = x * (1.0d0 - z)
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.2e+77) || !(x <= 4.4e+98)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.2e+77) or not (x <= 4.4e+98):
		tmp = x * (1.0 - z)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.2e+77) || !(x <= 4.4e+98))
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6.2e+77) || ~((x <= 4.4e+98)))
		tmp = x * (1.0 - z);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.2e+77], N[Not[LessEqual[x, 4.4e+98]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999997e77 or 4.40000000000000017e98 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. unsub-neg 89.9%

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

   5. Simplified 89.9%

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

   if -6.19999999999999997e77 < x < 4.40000000000000017e98

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 88.2%

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

      1. *-commutative 88.2%

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

   5. Simplified 88.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+77} \lor \neg \left(x \leq 4.4 \cdot 10^{+98}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+99}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+76)
   (* x (- 1.0 z))
   (if (<= x 4.3e+99) (+ x (* y z)) (- x (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+76) {
		tmp = x * (1.0 - z);
	} else if (x <= 4.3e+99) {
		tmp = x + (y * z);
	} else {
		tmp = x - (x * 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 <= (-3.1d+76)) then
        tmp = x * (1.0d0 - z)
    else if (x <= 4.3d+99) then
        tmp = x + (y * z)
    else
        tmp = x - (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+76) {
		tmp = x * (1.0 - z);
	} else if (x <= 4.3e+99) {
		tmp = x + (y * z);
	} else {
		tmp = x - (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e+76:
		tmp = x * (1.0 - z)
	elif x <= 4.3e+99:
		tmp = x + (y * z)
	else:
		tmp = x - (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+76)
		tmp = Float64(x * Float64(1.0 - z));
	elseif (x <= 4.3e+99)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = Float64(x - Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e+76)
		tmp = x * (1.0 - z);
	elseif (x <= 4.3e+99)
		tmp = x + (y * z);
	else
		tmp = x - (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e+76], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.3e+99], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e76

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. mul-1-neg 93.2%

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

      2. unsub-neg 93.2%

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

   5. Simplified 93.2%

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

   if -3.10000000000000011e76 < x < 4.3000000000000001e99

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 88.2%

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

      1. *-commutative 88.2%

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

   5. Simplified 88.2%

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

   if 4.3000000000000001e99 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in z around 0 87.0%

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

      1. mul-1-neg 87.0%

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

      2. sub-neg 87.0%

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

   8. Simplified 87.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+99}:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = x * -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 <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -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 <= -1.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. unsub-neg 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in z around inf 49.0%

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

      1. mul-1-neg 49.0%

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

      2. distribute-rgt-neg-out 49.0%

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

   8. Simplified 49.0%

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

   if -1 < z < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 71.4%

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

4. Final simplification 61.0%

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

Alternative 5: 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 6: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 z)))

C

double code(double x, double y, double z) {
	return x * (1.0 - 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 * (1.0d0 - z)
end function

Java

public static double code(double x, double y, double z) {
	return x * (1.0 - z);
}

Python

def code(x, y, z):
	return x * (1.0 - z)

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - z);
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 62.6%

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

   1. mul-1-neg 62.6%

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

   2. unsub-neg 62.6%

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

5. Simplified 62.6%

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

6. Final simplification 62.6%

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

Alternative 7: 37.2% 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 39.5%

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

4. Final simplification 39.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))