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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 7

Speedup: 1.2×

• 20.1% 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, 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: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-6} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e-6) (not (<= z 1.32e-145))) (* (- y x) z) (fma (- x) z x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-6) || !(z <= 1.32e-145)) {
		tmp = (y - x) * z;
	} else {
		tmp = fma(-x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e-6) || !(z <= 1.32e-145))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = fma(Float64(-x), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e-6], N[Not[LessEqual[z, 1.32e-145]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[((-x) * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999987e-6 or 1.32e-145 < 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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower-*.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -2.79999999999999987e-6 < z < 1.32e-145

   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. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.8

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

   7. Applied rewrites 78.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-6} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-6} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e-6) (not (<= z 1.32e-145))) (* (- y x) z) (* (- 1.0 z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-6) || !(z <= 1.32e-145)) {
		tmp = (y - x) * z;
	} else {
		tmp = (1.0 - z) * 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 <= (-2.8d-6)) .or. (.not. (z <= 1.32d-145))) then
        tmp = (y - x) * z
    else
        tmp = (1.0d0 - z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e-6) || !(z <= 1.32e-145)) {
		tmp = (y - x) * z;
	} else {
		tmp = (1.0 - z) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e-6) or not (z <= 1.32e-145):
		tmp = (y - x) * z
	else:
		tmp = (1.0 - z) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e-6) || !(z <= 1.32e-145))
		tmp = Float64(Float64(y - x) * z);
	else
		tmp = Float64(Float64(1.0 - z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.8e-6) || ~((z <= 1.32e-145)))
		tmp = (y - x) * z;
	else
		tmp = (1.0 - z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e-6], N[Not[LessEqual[z, 1.32e-145]], $MachinePrecision]], N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999987e-6 or 1.32e-145 < 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 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. lower-*.f64 N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -2.79999999999999987e-6 < z < 1.32e-145

   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 78.8

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

   5. Applied rewrites 78.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-6} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;\left(y - x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-83} \lor \neg \left(x \leq 3.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.6e-83) (not (<= x 3.9e-40))) (* (- 1.0 z) x) (* z y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-83) || !(x <= 3.9e-40)) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = z * y;
	}
	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 <= (-8.6d-83)) .or. (.not. (x <= 3.9d-40))) then
        tmp = (1.0d0 - z) * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-83) || !(x <= 3.9e-40)) {
		tmp = (1.0 - z) * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.6e-83) or not (x <= 3.9e-40):
		tmp = (1.0 - z) * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.6e-83) || !(x <= 3.9e-40))
		tmp = Float64(Float64(1.0 - z) * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.6e-83) || ~((x <= 3.9e-40)))
		tmp = (1.0 - z) * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.6e-83], N[Not[LessEqual[x, 3.9e-40]], $MachinePrecision]], N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision], N[(z * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.60000000000000066e-83 or 3.89999999999999981e-40 < x

   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -8.60000000000000066e-83 < x < 3.89999999999999981e-40

   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 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-83} \lor \neg \left(x \leq 3.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left(1 - z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-7} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e-7) (not (<= z 1.32e-145))) (* z y) (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-7) || !(z <= 1.32e-145)) {
		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 <= (-3.8d-7)) .or. (.not. (z <= 1.32d-145))) 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 <= -3.8e-7) || !(z <= 1.32e-145)) {
		tmp = z * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.8e-7) or not (z <= 1.32e-145):
		tmp = z * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e-7) || !(z <= 1.32e-145))
		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 <= -3.8e-7) || ~((z <= 1.32e-145)))
		tmp = z * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e-7], N[Not[LessEqual[z, 1.32e-145]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000015e-7 or 1.32e-145 < 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 52.4

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

   5. Applied rewrites 52.4%

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

   if -3.80000000000000015e-7 < z < 1.32e-145

   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 78.8

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-7} \lor \neg \left(z \leq 1.32 \cdot 10^{-145}\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\left(-z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-145}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0) (* (- z) x) (if (<= z 1.32e-145) (* 1.0 x) (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -z * x;
	} else if (z <= 1.32e-145) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	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 <= (-1.0d0)) then
        tmp = -z * x
    else if (z <= 1.32d-145) then
        tmp = 1.0d0 * x
    else
        tmp = z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = -z * x;
	} else if (z <= 1.32e-145) {
		tmp = 1.0 * x;
	} else {
		tmp = z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = -z * x
	elif z <= 1.32e-145:
		tmp = 1.0 * x
	else:
		tmp = z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(-z) * x);
	elseif (z <= 1.32e-145)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(z * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = -z * x;
	elseif (z <= 1.32e-145)
		tmp = 1.0 * x;
	else
		tmp = z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[((-z) * x), $MachinePrecision], If[LessEqual[z, 1.32e-145], N[(1.0 * x), $MachinePrecision], N[(z * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 58.4

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 56.9%

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

   if -1 < z < 1.32e-145

   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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.8%

      \[\leadsto 1 \cdot x \]

   if 1.32e-145 < 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 54.8

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

   5. Applied rewrites 54.8%

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

Alternative 7: 41.5% 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 40.9

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

5. Applied rewrites 40.9%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ x (* (- y x) z)))