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

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 6

Speedup: 1.0×

• 20.7% 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: 57.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-222}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-200}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.2e-109)
   (* y z)
   (if (<= z 6.5e-222)
     x
     (if (<= z 1.22e-200) (* y z) (if (<= z 2.2e-75) x (* y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.2e-109) {
		tmp = y * z;
	} else if (z <= 6.5e-222) {
		tmp = x;
	} else if (z <= 1.22e-200) {
		tmp = y * z;
	} else if (z <= 2.2e-75) {
		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 <= (-4.2d-109)) then
        tmp = y * z
    else if (z <= 6.5d-222) then
        tmp = x
    else if (z <= 1.22d-200) then
        tmp = y * z
    else if (z <= 2.2d-75) 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 <= -4.2e-109) {
		tmp = y * z;
	} else if (z <= 6.5e-222) {
		tmp = x;
	} else if (z <= 1.22e-200) {
		tmp = y * z;
	} else if (z <= 2.2e-75) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.2e-109:
		tmp = y * z
	elif z <= 6.5e-222:
		tmp = x
	elif z <= 1.22e-200:
		tmp = y * z
	elif z <= 2.2e-75:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.2e-109)
		tmp = Float64(y * z);
	elseif (z <= 6.5e-222)
		tmp = x;
	elseif (z <= 1.22e-200)
		tmp = Float64(y * z);
	elseif (z <= 2.2e-75)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.2e-109)
		tmp = y * z;
	elseif (z <= 6.5e-222)
		tmp = x;
	elseif (z <= 1.22e-200)
		tmp = y * z;
	elseif (z <= 2.2e-75)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.2e-109], N[(y * z), $MachinePrecision], If[LessEqual[z, 6.5e-222], x, If[LessEqual[z, 1.22e-200], N[(y * z), $MachinePrecision], If[LessEqual[z, 2.2e-75], x, N[(y * z), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999992e-109 or 6.5000000000000005e-222 < z < 1.22000000000000005e-200 or 2.20000000000000005e-75 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 67.6%

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

      1. *-commutative 67.6%

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

   4. Simplified 67.6%

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

   5. Taylor expanded in x around 0 59.1%

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

      1. *-commutative 59.1%

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

   7. Simplified 59.1%

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

   if -4.19999999999999992e-109 < z < 6.5000000000000005e-222 or 1.22000000000000005e-200 < z < 2.20000000000000005e-75

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 84.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-109}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-222}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-200}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-70} \lor \neg \left(x \leq 8.5 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.8e-70) (not (<= x 8.5e-140))) (* x (- 1.0 z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.8e-70) || !(x <= 8.5e-140)) {
		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 <= (-2.8d-70)) .or. (.not. (x <= 8.5d-140))) 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 <= -2.8e-70) || !(x <= 8.5e-140)) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.8e-70) or not (x <= 8.5e-140):
		tmp = x * (1.0 - z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.8e-70) || !(x <= 8.5e-140))
		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 <= -2.8e-70) || ~((x <= 8.5e-140)))
		tmp = x * (1.0 - z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e-70], N[Not[LessEqual[x, 8.5e-140]], $MachinePrecision]], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e-70 or 8.49999999999999997e-140 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 75.3%

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

      1. mul-1-neg 75.3%

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

   4. Simplified 75.3%

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

   5. Taylor expanded in x around 0 75.3%

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

   if -2.7999999999999999e-70 < x < 8.49999999999999997e-140

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 93.6%

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

      1. *-commutative 93.6%

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

   4. Simplified 93.6%

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

   5. Taylor expanded in x around 0 80.3%

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

      1. *-commutative 80.3%

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

   7. Simplified 80.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-70} \lor \neg \left(x \leq 8.5 \cdot 10^{-140}\right):\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-46} \lor \neg \left(y \leq 3.6 \cdot 10^{-75}\right):\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.4e-46) (not (<= y 3.6e-75))) (+ x (* y z)) (* x (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.4e-46) || !(y <= 3.6e-75)) {
		tmp = x + (y * 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 ((y <= (-7.4d-46)) .or. (.not. (y <= 3.6d-75))) then
        tmp = x + (y * 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 ((y <= -7.4e-46) || !(y <= 3.6e-75)) {
		tmp = x + (y * z);
	} else {
		tmp = x * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.4e-46) or not (y <= 3.6e-75):
		tmp = x + (y * z)
	else:
		tmp = x * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.4e-46) || !(y <= 3.6e-75))
		tmp = Float64(x + Float64(y * 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 ((y <= -7.4e-46) || ~((y <= 3.6e-75)))
		tmp = x + (y * z);
	else
		tmp = x * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.4e-46], N[Not[LessEqual[y, 3.6e-75]], $MachinePrecision]], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999966e-46 or 3.6e-75 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 89.7%

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

      1. *-commutative 89.7%

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

   4. Simplified 89.7%

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

   if -7.39999999999999966e-46 < y < 3.6e-75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 88.8%

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

      1. mul-1-neg 88.8%

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

   4. Simplified 88.8%

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

   5. Taylor expanded in x around 0 88.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-46} \lor \neg \left(y \leq 3.6 \cdot 10^{-75}\right):\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

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. Final simplification 100.0%

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

Alternative 6: 36.0% 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 32.8%

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

3. Final simplification 32.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))