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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 6

Speedup: 1.2×

• 20.4% 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

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

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: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -11500000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y x) z)))
   (if (<= z -11500000000.0) t_0 (if (<= z 1.0) (+ x (* y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -11500000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + (y * z);
	} 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 = (y - x) * z
    if (z <= (-11500000000.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + (y * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -11500000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + (y * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) * z
	tmp = 0
	if z <= -11500000000.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + (y * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) * z)
	tmp = 0.0
	if (z <= -11500000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + Float64(y * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) * z;
	tmp = 0.0;
	if (z <= -11500000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + (y * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -11500000000.0], t$95$0, If[LessEqual[z, 1.0], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e10 or 1 < 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 98.8

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

   5. Applied rewrites 98.8%

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

   if -1.15e10 < z < 1

   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 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -11500000000:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y - x\right) \cdot z\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-55}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- y x) z)))
   (if (<= z -2.2e-66) t_0 (if (<= z 9e-55) (- x (* x z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -2.2e-66) {
		tmp = t_0;
	} else if (z <= 9e-55) {
		tmp = x - (x * z);
	} 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 = (y - x) * z
    if (z <= (-2.2d-66)) then
        tmp = t_0
    else if (z <= 9d-55) then
        tmp = x - (x * z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) * z;
	double tmp;
	if (z <= -2.2e-66) {
		tmp = t_0;
	} else if (z <= 9e-55) {
		tmp = x - (x * z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) * z
	tmp = 0
	if z <= -2.2e-66:
		tmp = t_0
	elif z <= 9e-55:
		tmp = x - (x * z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) * z)
	tmp = 0.0
	if (z <= -2.2e-66)
		tmp = t_0;
	elseif (z <= 9e-55)
		tmp = Float64(x - Float64(x * z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) * z;
	tmp = 0.0;
	if (z <= -2.2e-66)
		tmp = t_0;
	elseif (z <= 9e-55)
		tmp = x - (x * z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -2.2e-66], t$95$0, If[LessEqual[z, 9e-55], N[(x - N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e-66 or 8.99999999999999941e-55 < z

   1. Initial program 99.9%

      \[x + \left(y - x\right) \cdot z \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if -2.2000000000000001e-66 < z < 8.99999999999999941e-55

   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. distribute-lft-out-- N/A

         \[\leadsto \color{blue}{x \cdot 1 - x \cdot z} \]

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-66}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-55}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.34:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 10^{+14}:\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.34) (* y z) (if (<= y 1e+14) (* z (- x)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.34) {
		tmp = y * z;
	} else if (y <= 1e+14) {
		tmp = z * -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 (y <= (-0.34d0)) then
        tmp = y * z
    else if (y <= 1d+14) then
        tmp = z * -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 (y <= -0.34) {
		tmp = y * z;
	} else if (y <= 1e+14) {
		tmp = z * -x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.34:
		tmp = y * z
	elif y <= 1e+14:
		tmp = z * -x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.34)
		tmp = Float64(y * z);
	elseif (y <= 1e+14)
		tmp = Float64(z * Float64(-x));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.34)
		tmp = y * z;
	elseif (y <= 1e+14)
		tmp = z * -x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.34], N[(y * z), $MachinePrecision], If[LessEqual[y, 1e+14], N[(z * (-x)), $MachinePrecision], N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.340000000000000024 or 1e14 < y

   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 67.4

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

   5. Applied rewrites 67.4%

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

   if -0.340000000000000024 < y < 1e14

   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 42.6

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

   5. Applied rewrites 42.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 36.4%

      \[\leadsto z \cdot \left(-x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 60.8

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

5. Applied rewrites 60.8%

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

6. Final simplification 60.8%

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

Alternative 6: 42.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y * 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 = y * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * z), $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 43.8

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

5. Applied rewrites 43.8%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))