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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 7

Speedup: 1.0×

• 21.1% 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. Final simplification 100.0%

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

Alternative 2: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-49}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= y -1.4e-64)
     (* y z)
     (if (<= y 2.15e-86)
       t_0
       (if (<= y 7e-49)
         (* y z)
         (if (<= y 3.8e-33) t_0 (if (<= y 3e+87) x (* y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (y <= -1.4e-64) {
		tmp = y * z;
	} else if (y <= 2.15e-86) {
		tmp = t_0;
	} else if (y <= 7e-49) {
		tmp = y * z;
	} else if (y <= 3.8e-33) {
		tmp = t_0;
	} else if (y <= 3e+87) {
		tmp = x;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (y <= (-1.4d-64)) then
        tmp = y * z
    else if (y <= 2.15d-86) then
        tmp = t_0
    else if (y <= 7d-49) then
        tmp = y * z
    else if (y <= 3.8d-33) then
        tmp = t_0
    else if (y <= 3d+87) then
        tmp = x
    else
        tmp = y * z
    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 (y <= -1.4e-64) {
		tmp = y * z;
	} else if (y <= 2.15e-86) {
		tmp = t_0;
	} else if (y <= 7e-49) {
		tmp = y * z;
	} else if (y <= 3.8e-33) {
		tmp = t_0;
	} else if (y <= 3e+87) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if y <= -1.4e-64:
		tmp = y * z
	elif y <= 2.15e-86:
		tmp = t_0
	elif y <= 7e-49:
		tmp = y * z
	elif y <= 3.8e-33:
		tmp = t_0
	elif y <= 3e+87:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (y <= -1.4e-64)
		tmp = Float64(y * z);
	elseif (y <= 2.15e-86)
		tmp = t_0;
	elseif (y <= 7e-49)
		tmp = Float64(y * z);
	elseif (y <= 3.8e-33)
		tmp = t_0;
	elseif (y <= 3e+87)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (y <= -1.4e-64)
		tmp = y * z;
	elseif (y <= 2.15e-86)
		tmp = t_0;
	elseif (y <= 7e-49)
		tmp = y * z;
	elseif (y <= 3.8e-33)
		tmp = t_0;
	elseif (y <= 3e+87)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[y, -1.4e-64], N[(y * z), $MachinePrecision], If[LessEqual[y, 2.15e-86], t$95$0, If[LessEqual[y, 7e-49], N[(y * z), $MachinePrecision], If[LessEqual[y, 3.8e-33], t$95$0, If[LessEqual[y, 3e+87], x, N[(y * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.40000000000000002e-64 or 2.15000000000000007e-86 < y < 7.00000000000000012e-49 or 2.9999999999999999e87 < y

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around 0 78.7%

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

   if -1.40000000000000002e-64 < y < 2.15000000000000007e-86 or 7.00000000000000012e-49 < y < 3.79999999999999994e-33

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 90.7%

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

   3. Taylor expanded in z around inf 58.5%

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

      1. associate-*r* 58.5%

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

      2. mul-1-neg 58.5%

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

   5. Simplified 58.5%

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

   if 3.79999999999999994e-33 < y < 2.9999999999999999e87

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 42.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-49}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-80} \lor \neg \left(z \leq 2.8 \cdot 10^{-28}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.2e-80) (not (<= z 2.8e-28))) (* (- y x) z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.2e-80) || !(z <= 2.8e-28)) {
		tmp = (y - 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 <= (-4.2d-80)) .or. (.not. (z <= 2.8d-28))) then
        tmp = (y - 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 <= -4.2e-80) || !(z <= 2.8e-28)) {
		tmp = (y - x) * z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.2e-80) or not (z <= 2.8e-28):
		tmp = (y - x) * z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.2e-80) || !(z <= 2.8e-28))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.2e-80) || ~((z <= 2.8e-28)))
		tmp = (y - x) * z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.2e-80], N[Not[LessEqual[z, 2.8e-28]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000003e-80 or 2.7999999999999998e-28 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around inf 97.6%

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

   if -4.20000000000000003e-80 < z < 2.7999999999999998e-28

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 67.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-80} \lor \neg \left(z \leq 2.8 \cdot 10^{-28}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+55}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e+55) (* y z) (if (<= y 3.15e+87) (* x (- 1.0 z)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e+55) {
		tmp = y * z;
	} else if (y <= 3.15e+87) {
		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 (y <= (-4.5d+55)) then
        tmp = y * z
    else if (y <= 3.15d+87) 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 (y <= -4.5e+55) {
		tmp = y * z;
	} else if (y <= 3.15e+87) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.5e+55:
		tmp = y * z
	elif y <= 3.15e+87:
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e+55)
		tmp = Float64(y * z);
	elseif (y <= 3.15e+87)
		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 (y <= -4.5e+55)
		tmp = y * z;
	elseif (y <= 3.15e+87)
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e+55], N[(y * z), $MachinePrecision], If[LessEqual[y, 3.15e+87], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.49999999999999998e55 or 3.15e87 < y

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around 0 86.1%

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

   if -4.49999999999999998e55 < y < 3.15e87

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around inf 79.7%

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

      1. distribute-rgt1-in 79.6%

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

      2. mul-1-neg 79.6%

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

      3. cancel-sign-sub-inv 79.6%

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

   4. Simplified 79.6%

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

   5. Taylor expanded in x around 0 79.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+55}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 5: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.2e-80) (* y z) (if (<= z 1.05e-26) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e-80) {
		tmp = y * z;
	} else if (z <= 1.05e-26) {
		tmp = x;
	} 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 (z <= (-2.2d-80)) then
        tmp = y * z
    else if (z <= 1.05d-26) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.2e-80) {
		tmp = y * z;
	} else if (z <= 1.05e-26) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.2e-80:
		tmp = y * z
	elif z <= 1.05e-26:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.2e-80)
		tmp = Float64(y * z);
	elseif (z <= 1.05e-26)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.2e-80)
		tmp = y * z;
	elseif (z <= 1.05e-26)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.2e-80], N[(y * z), $MachinePrecision], If[LessEqual[z, 1.05e-26], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e-80 or 1.05000000000000004e-26 < z

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in x around 0 54.3%

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

   if -2.2000000000000001e-80 < z < 1.05000000000000004e-26

   1. Initial program 100.0%

      \[x + \left(y - x\right) \cdot z \]

   2. Taylor expanded in z around 0 67.5%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

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. Final simplification 100.0%

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

Alternative 7: 37.0% 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. Taylor expanded in z around 0 26.6%

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

3. Final simplification 26.6%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))