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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 5

Speedup: 1.2×

• 21.4% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. *-rgt-identity N/A

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

   8. associate-+l+ N/A

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. distribute-rgt-out-- N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 100.0

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

5. Simplified 100.0%

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

Alternative 2: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -1.45e+215)
     (fma z y x)
     (if (<= z -1.4e+31) t_0 (if (<= z 5e+211) (fma z y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -1.45e+215) {
		tmp = fma(z, y, x);
	} else if (z <= -1.4e+31) {
		tmp = t_0;
	} else if (z <= 5e+211) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -1.45e+215)
		tmp = fma(z, y, x);
	elseif (z <= -1.4e+31)
		tmp = t_0;
	elseif (z <= 5e+211)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -1.45e+215], N[(z * y + x), $MachinePrecision], If[LessEqual[z, -1.4e+31], t$95$0, If[LessEqual[z, 5e+211], N[(z * y + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e215 or -1.40000000000000008e31 < z < 4.9999999999999995e211

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 85.9

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

   5. Simplified 85.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 85.9

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

   8. Simplified 85.9%

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

   if -1.45e215 < z < -1.40000000000000008e31 or 4.9999999999999995e211 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.8

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

   8. Simplified 75.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+215}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y - x\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- y x))))
   (if (<= z -1.0) t_0 (if (<= z 5e-8) (fma z y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y - x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= 5e-8) {
		tmp = fma(z, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y - x))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 5e-8)
		tmp = fma(z, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 5e-8], N[(z * y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 4.9999999999999998e-8 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 98.9

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

   5. Simplified 98.9%

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

   if -1 < z < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.5

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

   8. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 74.1

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

5. Simplified 74.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 74.1

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

8. Simplified 74.1%

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

Alternative 5: 41.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ z \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 39.5

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

5. Simplified 39.5%

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

6. Final simplification 39.5%

   \[\leadsto z \cdot y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))