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

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 5

Speedup: 1.2×

• 21.6% 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

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

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) \cdot x\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-10}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+51}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- z) x)))
   (if (<= z -4.2e+151)
     t_0
     (if (<= z -1.12e-10)
       (* z y)
       (if (<= z 4.4e-21) (* 1.0 x) (if (<= z 1.85e+51) (* z y) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -4.2e+151) {
		tmp = t_0;
	} else if (z <= -1.12e-10) {
		tmp = z * y;
	} else if (z <= 4.4e-21) {
		tmp = 1.0 * x;
	} else if (z <= 1.85e+51) {
		tmp = z * y;
	} 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 = -z * x
    if (z <= (-4.2d+151)) then
        tmp = t_0
    else if (z <= (-1.12d-10)) then
        tmp = z * y
    else if (z <= 4.4d-21) then
        tmp = 1.0d0 * x
    else if (z <= 1.85d+51) then
        tmp = z * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z * x;
	double tmp;
	if (z <= -4.2e+151) {
		tmp = t_0;
	} else if (z <= -1.12e-10) {
		tmp = z * y;
	} else if (z <= 4.4e-21) {
		tmp = 1.0 * x;
	} else if (z <= 1.85e+51) {
		tmp = z * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z * x
	tmp = 0
	if z <= -4.2e+151:
		tmp = t_0
	elif z <= -1.12e-10:
		tmp = z * y
	elif z <= 4.4e-21:
		tmp = 1.0 * x
	elif z <= 1.85e+51:
		tmp = z * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) * x)
	tmp = 0.0
	if (z <= -4.2e+151)
		tmp = t_0;
	elseif (z <= -1.12e-10)
		tmp = Float64(z * y);
	elseif (z <= 4.4e-21)
		tmp = Float64(1.0 * x);
	elseif (z <= 1.85e+51)
		tmp = Float64(z * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z * x;
	tmp = 0.0;
	if (z <= -4.2e+151)
		tmp = t_0;
	elseif (z <= -1.12e-10)
		tmp = z * y;
	elseif (z <= 4.4e-21)
		tmp = 1.0 * x;
	elseif (z <= 1.85e+51)
		tmp = z * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) * x), $MachinePrecision]}, If[LessEqual[z, -4.2e+151], t$95$0, If[LessEqual[z, -1.12e-10], N[(z * y), $MachinePrecision], If[LessEqual[z, 4.4e-21], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 1.85e+51], N[(z * y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2000000000000001e151 or 1.8500000000000001e51 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 65.1

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 65.1%

      \[\leadsto \left(-z\right) \cdot x \]

   if -4.2000000000000001e151 < z < -1.12e-10 or 4.4000000000000001e-21 < z < 1.8500000000000001e51

   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. *-commutative N/A

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

      2. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if -1.12e-10 < z < 4.4000000000000001e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-10} \lor \neg \left(z \leq 4.4 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.12e-10) (not (<= z 4.4e-21))) (* z (- y x)) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e-10) || !(z <= 4.4e-21)) {
		tmp = z * (y - x);
	} else {
		tmp = 1.0 * 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 <= (-1.12d-10)) .or. (.not. (z <= 4.4d-21))) then
        tmp = z * (y - x)
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e-10) || !(z <= 4.4e-21)) {
		tmp = z * (y - x);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.12e-10) or not (z <= 4.4e-21):
		tmp = z * (y - x)
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.12e-10) || !(z <= 4.4e-21))
		tmp = Float64(z * Float64(y - x));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.12e-10) || ~((z <= 4.4e-21)))
		tmp = z * (y - x);
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.12e-10], N[Not[LessEqual[z, 4.4e-21]], $MachinePrecision]], N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e-10 or 4.4000000000000001e-21 < z

   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. *-commutative N/A

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

      2. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 97.3

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

   8. Applied rewrites 97.3%

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

   if -1.12e-10 < z < 4.4000000000000001e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-10} \lor \neg \left(z \leq 4.4 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-10} \lor \neg \left(z \leq 4.4 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.12e-10) (not (<= z 4.4e-21))) (* z y) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e-10) || !(z <= 4.4e-21)) {
		tmp = z * y;
	} else {
		tmp = 1.0 * 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 <= (-1.12d-10)) .or. (.not. (z <= 4.4d-21))) then
        tmp = z * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.12e-10) || !(z <= 4.4e-21)) {
		tmp = z * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.12e-10) or not (z <= 4.4e-21):
		tmp = z * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.12e-10) || !(z <= 4.4e-21))
		tmp = Float64(z * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.12e-10) || ~((z <= 4.4e-21)))
		tmp = z * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.12e-10], N[Not[LessEqual[z, 4.4e-21]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e-10 or 4.4000000000000001e-21 < z

   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. *-commutative N/A

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

      2. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -1.12e-10 < z < 4.4000000000000001e-21

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-10} \lor \neg \left(z \leq 4.4 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% 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. *-commutative N/A

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

   2. lower-*.f64 42.7

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

5. Applied rewrites 42.7%

   \[\leadsto \color{blue}{z \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))